Abstract
We give sufficient and necessary conditions for the non-emptiness of the alpha-core in the setting of strategic games with non-ordered and discontinuous preferences. In order to prove our results, we can avoid the use of Scarf’s Theorem for NTU-games, by suitably appealing to the Ky Fan minimax inequality. Examples clarify our conditions and allow the comparison of our results with the previous ones.
Highlights
A well known cooperative solution concept for strategic form games is the α-core by Aumann (1961)
Remark 4.2 Theorem 4.1 presents a class of discontinuous games where the non-emptiness of the α-core is characterized by means of a quasi-concavity like property
Inspired by a recent literature on the existence of Nash equilibria in games with non-ordered preferences, our results have been obtained by means of the Ky Fan minimax inequality
Summary
A well known cooperative solution concept for strategic form games is the α-core by Aumann (1961). We apply the general results on the existence of solutions to the Ky Fan minimax inequality given by Scalzo (2013, 2019a) and identify sufficient (and necessary) conditions on the game for the non-emptiness of the α-core. The aggregate preference of a club Ci may not satisfy the continuity properties considered by the previous literature (open-graph preferences). This is the case, for example, if i is interpreted as the administrator of a condominium Ci. By the role, she/he is naturally led to identify amounts even when they are different but differ by less than a threshold level λ > 0.
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